Relative Minimum In Calculus With First And Second Derivative Test

relative/local minimum

-at c if f(c) is less than or equal to f(x) for all x in a neighborhood of c

A relative (or local) minimum is a point on a curve where the function has a lower value than at any neighboring points, but this minimum is not necessarily the absolute minimum of the function within its domain. In other words, a relative minimum is a point where the function is lower than the values nearby, but it may not be the lowest point on the entire curve.

To determine a relative minimum, we can use the first derivative test or the second derivative test.

The first derivative test involves finding the critical points of the function by solving for the derivative of the function equalling zero. Then, we check the sign of the derivative on either side of the critical point to determine if it is a maximum, minimum, or neither. If the derivative changes from positive to negative at the critical point, then it is a relative maximum. If it changes from negative to positive, then it is a relative minimum.

The second derivative test involves finding the critical points of the function and then taking the second derivative. We then determine the sign of the second derivative at the critical point. If it is positive, then it is a relative minimum. If it is negative, then it is a relative maximum. If the second derivative is zero, then the test is inconclusive and we need to use the first derivative test.

It is important to note that a relative minimum is only a minimum within a small interval around the point. There may be lower points on the curve outside of this interval.

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